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The book aims to expose graduate students and young researchers to a range of advanced topics in gravitation, while presenting the calculations in as explicit and detailed a manner as possible.The first volume begins by examining the consequences of diffeomorphism invariance for solutions of the Einstein field equations, illustrated through various formulations of the Schwarzschild solution. This is followed by a detailed construction of Carter-Penrose conformal diagrams, together with remarks highlighting differences from the presentations commonly found in standard textbooks.The chapter on geodesics and congruences introduces the spacetime splittings of Ellis, Clarkson, and Hayward, as well as kinematic and dynamical approaches based on optical scalars—tools that are pervasive in cosmology, horizon studies, gravitational lensing, and gravitational-wave physics. This chapter also includes a section on the most commonly used energy conditions and on fluid forms of the energy-momentum tensor.Subsequently, following a discussion of horizons, trapping surfaces, and the surface gravity of Killing horizons, the book presents a detailed exposition of several singularity theorems, including their key concepts and selected proofs. The chapter on the Kerr solution addresses issues of conserved charges and integration in gravity, invariants, and the global properties of the spacetime, including horizons and singularities, as well as its conformal diagram and geodesic structure. This naturally leads to the area theorem and black hole thermodynamics.The discussion then proceeds to relativistic thermodynamics, with a chapter devoted to the Eckart and Israel-Stewart formalisms. The book concludes with two technical chapters: one on the analytical mechanics approach to geodesics using the Hamilton-Jacobi formalism, and another on the Mathisson-Papapetrou-Dixon equations describing the motion of spinning bodies in general relativity.
